PROJECTION PROBLEM FOR REGULAR POLYGONS

Transcription

1 Joural of Mathematcal Sceces: Advaces ad Applcatos Volume, Number, 008, Pages PROJECTION PROBLEM FOR REGULAR POLYGONS College of Scece Bejg Forestry Uversty Bejg 0008 P. R. Cha e-mal: Abstract I ths paper a questo posed about a affe varat fuctoal defed by Lutwak et al. [5] s studed ad the aswer for regular polygos s gve.. Itroducto If K s a covex body (a compact, covex subset wth oempty teror) -dmesoal Eucldea space R, the ts support fucto h( K, ) : S R s defed by where body h( K, u) max{ u x : x K}, u x deotes the stadard er product of u ad x. The projecto K of K ca be defed as the covex body whose support fucto, for u S, s gve by 000 Mathematcs Subject Classfcato: 5A0. Keywords ad phrases: regular polygo, Scheder s projecto problem, affe varat fuctoal, Joh posto. Supported by Natoal Natural Sceces Foudato of Cha (077). Receved February, 008; Revsed Aprl 7, Scetfc Advaces Publshers

2 9 ( K u) vol ( K u ), h, where vol deotes ( ) -dmesoal volume ad K u deotes the mage of the orthogoal projecto of K oto the codmeso subspace orthogoal to u. For more formato about covex bodes, see the books of Garder [] ad Scheder [7]. A mportat usolved problem s Scheder s projecto problem [, 7, 8]: What s the least upper boud, as K rages over the class of org-symmetrc covex bodes, of the affe-varat rato [ V ( K ) V ( K ) ], where V s used to abbrevate vol. Just as metoed [5] 98, Scheder [] cojectured that ths rato s maxmzed by parallelotopes. I [], Scheder also preseted applcatos of such results stochastc geometry. However, 99, a couterexample was produced [] to show ths s ot the case. I 00, Lutwak et al. [5] troduced the followg ew cetro-affe fuctoal for a covex polytope ad preseted a modfed verso of Scheder s projecto problem. Defto. If P s a covex polytope R whch cotas the org ts teror, ad u,, un are the outer ormal ut vectors to the faces of P, wth h,, hn the correspodg dstaces of the faces from the org ad a,, an the correspodg areas of the faces, the defe U ( P ) by U( P ) h h. a a u 0 u Based o the above defto, Lutwak et al. proved the followg theorem.

3 PROJECTION PROBLEM FOR REGULAR POLYGONS 97 Theorem. If K s a org-symmetrc covex polytope R, the V ( K ) U( K ) V ( K )! wth equalty f ad oly f K s a parallelotope. Further, they posed two questos. Questo A. If P s a org-symmetrc covex polytope R, the s t the case that U( P ) ( ) V ( P ), wth equalty f ad oly f P s a parallelotope? Recetly, He et al. [] resolved Questo A the affrmatve. The Joh ellpsod of a covex body s the largest ( volume) ellpsod that s cotaed the body. The Joh pot of a covex body s the ceter of the Joh ellpsod of the body. A covex body R s sad to be Joh posto f ts Joh ellpsod s the stadard ut ball R. The other questo posed by Lutwak et al. [5] s:!, Questo B. Suppose P s a covex polytope pot at the org. Is t the case that R wth ts Joh U( P ) [( + )!] V ( P ), + wth equalty f ad oly f P s a smplex? I ths paper, we gve a partal aswer to Questo B R, obtag the followg theorem. Theorem. If P s a regular -polygo the org, the for R wth ts Joh pot at

4 98 U( P ) V ( P ), wth equalty f ad oly f P s a regular hexago. For two covex bodes K, L R, the Hausdorff metrc δ ( K, L) s defed by δ( K, L) sup h( K, u) h( L, u). u S Note that the fuctoal U s cetro-affe varat that, U ( φ P ) U( P ), for all φ SL( ) ad all regular polygos P wth Joh pot at the org. Deote the set of regular polygo polygos R by R by P. Defe the set C of covex C { K ε > 0, P P ad φ SL( ) such that δ( K, φp ) < ε }. The we ca get the followg corollary from Theorem. Corollary. If K C, the for U( K ) V ( K ).. The Proof of the Ma Result The well-kow theorem of Joh characterzes the ellpsod of maxmal volume cotaed a covex body ad s stated as a lemma. See []. Lemma. Let C be a covex body R. The ellpsod of maxmal volume C s B, f ad oly f C cotas ) m ) m pots ( u ad postve umber ( c so that m (a) c u u I, ad B ad there are cotact

5 PROJECTION PROBLEM FOR REGULAR POLYGONS 99 m (b) c u 0. ) m Codto (a) shows that the ( u behave rather lke a orthoormal bass that we ca resolve the Eucldea orm as a (weghted) sum of squares of er products. Ths codto s equvalet to the statemet that, for all x R, x m c x, u, where, s the usual Eucldea er product ad s the duced orm by ths er product. At frst, we wll show that the scrbed crcle of regular polygo s ts Joh crcle by the method dfferet from the usual affe trasformato. For ths, oe eed oly prove that the taget pots of the regular polygo wth ts scrbed crcle satsfy the codto (a) ad (b) of Lemma. Wthout loss of geeralzato, we suppose that a regular polygo P cotas the ut crcle B as ts scrbed crcle wth u, u,, u arrayed atclockwse as the ut outer ormal vectors of the sdes of P. Suppose that z, z,, z are the complex umbers correspodg to u, u,, u. The a equvalet codto to codto (b) s that c z 0, for some c. Suppose that u agrees wth the ut vector o the x-axs ad the agle betwee u ad u + s θ. The z cos 0 + s 0, z cos( ) θ + s( ) θ,,,,.

6 500 Observe that z, z,, z are the roots of the complex equato Thus by Vete s theorem we obta z 0. z + z + + z 0. Let c,,,,. The Thus codto (b) s satsfed. c z 0. Next, we wll prove that the u satsfy codto (a) of Lemma, that s, for ay x R, the followg equalty holds: x m c x, u, () where c,,,. Equalty () s equvalet to u, u, () for all u S. Let that u ( cos ϕ, s ϕ) ad ote that ( cos( ) θ, s( ) θ), because u (,,, ) correspod to the roots of z. We the have u u, u [ cos( ) θ cos ϕ + s( ) θ s ϕ] cos [( ) θ + ϕ]

7 PROJECTION PROBLEM FOR REGULAR POLYGONS 50 cos[ ( ) θ + ϕ] + + cos[ ( ) θ + ϕ]. Note that G { z cos( ) θ + s( ) θ, (, ), } s a cyclc group. If we deote cos[ ( ) θ] + s[ ( ) θ] (,, ), the z G { z, (,, ), } s a cyclc subgroup of G. Thus there must exst a complex equato ( z ) such that z are the roots. By Vete s theorem, we have cos[ ( ) θ + ϕ] cos[ ( ) θ] 0. Thus codto (a) s satsfed ad we have the cocluso that the scrbed crcle of regular polygo s ts Joh crcle. The followg lemma s also eeded the proof of Theorem. Lemma. If P s a polygo wth sdes, the the par umber of mutually parallel sdes s o more tha. Proof. Case. For,, 5, the result s obvous. Case. For, suppose that a s a sde of P wth outer ut ormal vector u. The the umber of sdes parallel to a s o more tha oe. I fact, f ot, suppose that a, a, j k wth outer ut ormal vectors u u, respectvely, are both parallel to a. The the j, k agle θ j betwee u ad u j s π ad the agle θ k betwee u ad u k s also π. Ths cotradcts the fact that θj + θk < π. Thus we get that f P s a polygo wth sdes, the the par umber of mutually parallel sdes s o more tha. j k

8 50 Proof of Theorem. By defto of U ( P ) ad Lemma, we have U( P ) h h a a u u 0 a j a u u 0 ( ) a a ( ) a a a ( ). By drectly computg, we have Thus s equvalet to or V ( P ) a. U( P ) V ( P ), ( ) a, a 0. So for we have ad U( P ) V ( P ). For whe P s a regular hexago, we have U ( P ) ( a + + a ) a

9 PROJECTION PROBLEM FOR REGULAR POLYGONS 50 a V ( P ) 9, so U ( P ) V ( P ). Remark ad so () For whe P s a regular tragle, we have U ( P ) ( a + a + a + a + a + a ) a 9 a V ( P ), U ( P ) V ( P ). () For, by drectly computg, we have ad U ( P ) ( a + a + a + a + a + a + a + a ) a a V ( P ), so U ( P ) < V ( P ). Ths s a couterexample to Questo B R. ad () For 5, by drectly computg, we have U ( P ) ( a + + a ) 5a 0

10 50 5 V ( P ) a, so U ( P ) > V ( P ). Ackowledgmet The author thaks the referee for may valuable suggestos. Refereces [] K. Ball, A elemetary troducto to moder covex geometry, Flavors of Geometry, S. Levy, ed., Cambrdge Uversty Press, New York, 997, pp [] N. S. Brae, Volumes of projecto bodes, Mathematka (99), 55-. [] R. J. Garder, Geometrc Tomogrophy, Cambrdge Uversty Press, Cambrdge, 995. [] Bwu He, Gagsog Leg ad Kagha L, Projecto problems for symmetrc polytopes, Adv. Math. 07 (00), [5] E. Lutwak, D. Yag ad G. Zhag, A ew affe varat for polytopes ad Scheder s projecto problem, Tras. Amer. Math. Soc. 5 (00), [] R. Scheder, Radom hyperplaes meetg a covex body, Z. Wahrschelchketsth. Verw. Geb. (98), [7] R. Scheder, Covex Bodes: The Bru-Mkowsk Theory, Cambrdge Uversty Press, Cambrdge, 99. [8] R. Scheder ad W. Wel, Zoods ad related topcs, Covexty ad ts Applcatos, P. M. Gruber ad J. M. Wlls, eds., Brkhäuser, Basel, 98, pp g